Generally, within a similar group of animals, for example, mammals, metabolic rate is higher for smaller body size versus larger body size.

You have to take into consideration the mass (body size) of the animal though. that is why its called the mass specific metabolic rate (MSMR).

For your example, a rat has smaller body size but it's MSMR is much higher than that of the elephant (larger body size). The surface area to volume ratio (SA:V) of a rat is larger than that of the elephant. you can prove this with math and geometry.

one of the ideas is that because the rat is so small, but its surface area ( the area covered by skin and fur) to volume ratio is large, it can loose a lot more heat. in order to survive and keep warm, it needs to rev up its metabolic rate to compensate for that heat loss. of course there are other things involved (like type of insulation such as fat, fur type and length, whether they are of another groups like birds, or lizards, etc). but in general this SA:V and MSMR hold true.